Nonuniform oversampled filter banks for audio signal processing

ABSTRACT

A non-uniform filter bank is created by joining sections of oversampled uniform filter bands that are based on complex exponential modulation (as opposed to cosine modulation). Each filter bank handles a given, non-overlapping frequency band. The bands are not of uniform bandwidth, and the filters of different banks have different bandwidths. The frequency bands of the different filter banks cover the frequency of interest with gaps in the neighborhoods of the filter band edges. A set of transition filters fills those gaps.

BACKGROUND OF THE INVENTION

This application is a continuation of U.S. patent application Ser. No.11/233,322, filed Sep. 22, 2005, which is a continuation of U.S. Pat.No. 6,996,198 issued Feb. 7, 2006, and it claims priority underprovisional application No. 60/243,811 filed on Oct. 27, 2000.

This invention relates to filters and, more particularly, to filterbanks for audio applications.

In many areas of audio processing, it is desirable to analyze an audiosignal in approximately the same time-frequency form as the human ear(i.e., with bandwidths on the order of one Bark) and with a timeresolution that is commensurate with the bandwidth of the filter. Inaddition, it is desirable to process the signals in each of the bandsand then reconstruct them in a manner such that when the bands areunmodified, the filter bank has a nearly perfect reconstructioncharacteristic. Because the signals might be modified, and differentbands might be routed to different devices, not only must the filtersprovide approximately exact reconstruction, they must also preventaliasing due to the unequal processing, or modification, of adjacentfrequency bands. Hence, an oversampled filter bank is required wherealiasing introduced due to unequal processing of bands is below thelevel of human hearing.

One application for this kind of filter bank is found for the problem ofseparating parts of an individual audio signal into its directs andindirect parts for the purpose of rerouting, in real time, the directand indirect signals to drivers that reproduce them appropriately. Insuch an application, a filter bank that approximates the criticalbandwidths is essential to identifying the part of a signal with directcues, and the ability of the reconstruction filter bank to preventsubstantial aliasing when adjacent bands are added incoherently (asopposed to coherently) is also an absolute requirement. Hence the needfor an oversampled critical band filter bank. In applications thatrequire nonuniform division of signal spectrum, iterative cascaded ofuniform filter banks are often used. Iterated filter banks, however,impose considerable structure on the equivalent filters, which resultsin inferior time-frequency localization compared to direct designs. Astudy of critically sampled nonuniform filter banks has been reported byJ. Princen in “The Design of Nonuniform Filter Banks,” IEEE Transactionson Signal Processing, Vol. 43, No. 11, pp. 2550-2560, November 1995.Nonuniform filter banks studied by Princen are obtained by joiningpseudo QMF filter bank sections that are nearly perfect reconstructionfilter banks based on cosine modulation and the principle of adjacentchannel aliasing cancellation. R. Bernardini et al published “ArbitraryTilings of the Time-Frequency Plane using Local Bases,” IEEETransactions on Signal Processing, Vol. 47, No. 8, pp. 2293-2304, August1999, wherein they describe a cosine-modulation-based structure thatallows for time-adaptive nonuniform tiling of the time-frequency plane.Despite their many fine features that are relevant to coding purposes,however, these approaches do not have good aliasing attenuationproperties in each of the subbands independently. This makes themunsuitable for tasks where processing effects need to be containedwithin the bands directly affected. Perfect, or nearly perfect,reconstruction properties of these filter banks in the presence ofupsampling are also not clear. The pseudo QMF bank, for instance, losesits aliasing cancellation property if the subband components are notcritically downsampled.

Oversampled uniform filter banks based on cosine modulation were studiedby Bolceskei et al, and reported in “Oversampled Cosine Modulated FilterBanks with Perfect Reconstruction, IEEE Transaction on Circuits andSystems II, Vol. 45, No. 8, pp. 1057-1071, August 1998, but the cosinemodulation places stringent aliasing attenuation requirements.

SUMMARY OF THE INVENTION

An advance in the art is attained with non-uniform filter banks createdby joining sections of oversampled uniform filter banks that are basedon complex exponential modulation (as opposed to cosine modulation).Each filter bank handles a given, non-overlapping frequency band, andthe filters of different banks have different bandwidths. The frequencybands of the different filter banks cover the frequency of interest withgaps in the neighborhoods of the filter band edges. A set of transitionfilters fills those gaps.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 shows an analysis filter bank stage, followed by a signalprocessing stage and a synthesis filter bank stage;

FIG. 2 shows the FIG. 1 arrangement, modified with a subsampling elementinterposed between the output of each analysis filter bank and thesignal processing stage, and an upsampling element interposed betweenthe signal processing stage and each synthesis filter bank;

FIG. 3 shows the frequency response of the filter arrangement disclosedherein; and

FIG. 4 presents a block diagram of the filter arrangement disclosedherein.

DETAILED DESCRIPTION

In accordance with the principles disclosed herein, nonuniform,oversampled filter banks are obtained by joining section of differentuniform filter banks with the aid of transition filters. The uniformfilter banks that are used are nearly perfect-reconstruction,oversampled, modulated filter banks.

FIG. 1 shows an analysis filter bank 100, followed by a processing stage200, and a synthesis filter bank 300. Analysis filter bank 100 is shownwith N filter elements (of which elements 11, 12, 13 are explicitlyshown), where each is a modulated version of a given window functionν[n]. That is, the time response of the i^(th) filter in the bank is:

$\begin{matrix}\begin{matrix}{{h_{i}\lbrack n\rbrack} = {{v\lbrack n\rbrack}{\mathbb{e}}^{j\frac{2\pi}{N}{({i - 1})}n}}} & {{i = 1},2,\cdots\mspace{14mu},{N.}}\end{matrix} & (1)\end{matrix}$The term

${\mathbb{e}}^{j\frac{2\pi}{N}{({i - 1})}n}$creates filters that are offset in frequency from one another, through acomplex modulation of the fundamental filter function ν[n], and someartisans call these filters “modulates” of the filter ν[n].

It can be shown that, when there is no subsampling in the channels ofsuch a filter bank (and, indeed, analysis filter 100 shows nosubsampling), a necessary and sufficient condition for perfect andnumerically stable reconstruction of any signal, x, from its subbandcomponents y_(i), where

$\begin{matrix}{{y_{i} = {\sum\limits_{k}\;{{x\lbrack k\rbrack}{h_{i}\left\lbrack {n - k} \right\rbrack}}}},} & (2)\end{matrix}$is given by

$\begin{matrix}{{0 < A \leq {\sum\limits_{i = 0}^{N - 1}\;{{V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N}i}})}} \right)}}^{2}} \leq B < \infty},{\omega \in {\left( {{- \pi},\pi} \right).}}} & (3)\end{matrix}$

In accord with the principles disclosed herein, however, a subclass ofwindows is considered that satisfy the equation (3) condition with thecaveat that A=B=N; which reduces equation (3) to:

$\begin{matrix}{{\sum\limits_{i = 0}^{N - 1}\;{{V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N}i}})}} \right)}}^{2}} = {{N\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}\omega} \in {\left( {{- \pi},\pi} \right).}}} & (4)\end{matrix}$The power complementarity condition of equation (4) permits good controlover the effects of processing in the subband domain, and also assuresstability.

When the analysis filters satisfy this condition, the norm of the inputsignal is related to the norm of the corresponding subband componentsy_(i) as follows:

$\begin{matrix}{{x}^{2} \leq {\frac{1}{N}{\sum\limits_{i}\;{{y_{i}}^{2}.}}}} & (5)\end{matrix}$It is noted that the considered signals belong to the space of squaresummable sequences, so the norm of x corresponds to

${x} = {\left( {\sum\limits_{n}\;{{x\lbrack n\rbrack}}^{2}} \right)^{\frac{1}{2}}.}$

If some processing modifies the subband components y_(i) to y_(i)+e_(i)(that is, the input signals to filter bank 300 in FIG. 1 are the signals(y_(i)+e_(i)), the total distortion in the signal x that is synthesizedfrom the modified subband components is bounded by

$\begin{matrix}{{e_{x}}^{2} \leq {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}\;{{e_{i}}^{2}.}}}} & (6)\end{matrix}$On the other hand, if the window ν satisfies the looser form of theperfect reconstruction condition given by equation (3), the distortionsin the subband components and the distortion in the synthesized signalare related as

$\begin{matrix}{{\frac{1}{B}{\sum\limits_{i = 0}^{N - 1}\;{e_{i}}^{2}}} \leq {e_{x}}^{2} \leq {\frac{1}{A}{\sum\limits_{i = 0}^{N - 1}\;{{e_{i}}^{2}.}}}} & (7)\end{matrix}$Equation (7) indicates that the distortion in the synthesized signal maygrow considerably out of proportion when A is small. Thus, thedistortion limit of equation (6) is one advantage that arises fromadopting the power complementarity condition of equation (4). Anotheradvantage that arises from adopting the power complementarity conditionof equation (4) is that an input signal can be perfectly reconstructedusing a synthesis filter bank that consists of filters

${\overset{\sim}{V}\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N}i}})}} \right)},$which are time-reversed versions of the analysis filters

${V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N}i}})}} \right)}.$It is such filters that are depicted in filter bank 300 of FIG. 1 (withdesignation V(z) corresponding to the time-reversed version of filterV(z)). That provides the convenience of not having to deal with thedesign of a synthesis window. Thus, the synthesis filter shown in FIG. 1depicts filters 31, 32, and 33 that are the time-reversed versions ofcorresponding analysis filters 11, 12, and 13. It may be noted that thepower complementarity condition of equation (4) also holds for thesynthesis filter; i.e.,

${\sum\limits_{i = 0}^{N - 1}\;{{V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N}i}})}} \right)}}^{2}} = {{N\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}\omega} \in {\left( {{- \pi},\pi} \right).}}$

FIG. 2 shows an analysis filter bank where, following each of thefilters there is an associated subsampling circuit. That is, circuits21, 22, and 23 follow filters 11, 12, and 13, respectively.Correspondingly, in the synthesis filter bank there are upsamplingcircuits 24, 25 and 26 that respectively precede filters 31, 32, and 33.In the case of the FIG. 2 arrangement where there is subsampling by K inthe analysis channels, and the input signal is reconstructed from thesubband components using time-reversed versions of the analysis filterspreceded by K-factor upsampling, the reconstructed signal, in theFourier domain, is given by

$\begin{matrix}{{X_{r}\left( {\mathbb{e}}^{j\omega} \right)} = {{\frac{1}{K}{\sum\limits_{i = 0}^{N - 1}\;{{V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N}i}})}} \right)}}^{2}}} + {\frac{1}{K}{\sum\limits_{k = 1}^{K - 1}\;{{X\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{K}k}})}} \right)}{A_{k}\left( {\mathbb{e}}^{j\omega} \right)}}}}}} & (8)\end{matrix}$where A_(k)(e^(jω)) are aliasing components

$\begin{matrix}{{A_{k}\left( {\mathbb{e}}^{j\omega} \right)} = {\sum\limits_{i = 0}^{N - 1}\;{{V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N}i}})}} \right)}{{V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N}i} - {\frac{2\pi}{K}k}})}} \right)}.}}}} & (9)\end{matrix}$Based on M. R. Portnoff, “Time-Frequency Representation of DigitalSignals and Systems Based on Short-Time Fourier Analysis,” IEEETransactions on Acoustics Speech and Signal Processing, Vol. 28, No. 1,pp. 55-69, February 1980; and Z. Cvetkovic, “On Discrete Short-TimerFourier Analysis,” IEEE Transaction on Signal Processing, Vol. 48, No.9, pp. 2628-2640, September 2000, it can be shown that the aliasingcomponents reduce to zero, and a perfect reconstruction condition isattained, if the window satisfies the constraint:

$\begin{matrix}{{{\sum\limits_{j}\;{{v\left\lbrack {k + {jK}} \right\rbrack}{v\left\lbrack {k^{\prime} + {iN} + {jK}} \right\rbrack}}} = {\frac{1}{K}{\delta\lbrack i\rbrack}}},{k = 0},1,{{\cdots\mspace{14mu} K} - 1}} & (10)\end{matrix}$where

$\begin{matrix}\begin{matrix}{{\delta\lbrack i\rbrack} = 1} & {{{when}\mspace{14mu} i} = 0} \\{\mspace{40mu}{= 0}} & {otherwise}\end{matrix} & (11)\end{matrix}$

As indicated above, the windows that are considered herein are thosethat satisfy the power complementarity condition in equation (4), andprovide a nearly perfect reconstruction by having high enoughattenuation for ω>2π/K. That is, there is no significant aliasingcontribution due to subsampling; i.e.,

$\begin{matrix}{{{V\left( {\mathbb{e}}^{j\omega} \right)}{V\left( {\mathbb{e}}^{{j\omega} - \frac{2\pi}{K}} \right)}} \approx {0\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}{\omega.}}} & (12)\end{matrix}$This makes aliasing sufficiently low in each subband independently.

In order to facilitate design of transition filters, it is convenient todeal with windows that also have sufficiently high attenuation forfrequencies ω>2π/N. Sufficiently high attenuation for that purposesmeans that in the power complementarity formula of equation (4), for any2πi/N<ω<2π(i+1)/N, the only significant contribution to equation (4)comes from a filter having a center frequency at 2πi/N and the filterhaving a center frequency at 2π(i+1)/N, the contribution of otherfilters is negligible. Stated in other words, a filter spills energyinto the band of no other filters except, possibly, the immediatelyadjacent filters. Stated in still other words, the attenuation of afilter centered at 2πi/N at frequencies (2π(i−1))/N>ω>(2π(i+1))/N isgreater than a selected value.

The above addresses uniform filters, with subsampling of K. One aspectof the arrangement disclosed herein, however, is the use of filters thatare not necessarily of uniform bandwidth, where a signal can be analyzedwith a time resolution that is commensurate with the bandwidth of thefilters, and have these filters be such that a nearly perfectreconstruction of the signal is possible. Consider, therefore, a firstfilter section, such as section 100 in FIG. 2, with subsampling of K₁,and a window function

${v_{1}\left( {V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N_{1}}i}})}} \right)} \right)},$and a second filter section, such as section 100 in FIG. 2, withupsampling of K₂, and a window function

${v_{2}\left( {V\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N_{2}}i}})}} \right)} \right)}.$Consider further that window ν₁ filters are employed below the“break-over” frequency ω₀ and windows ν₂ are employed above ω₀.Consequently, there is a gap in the frequency response function, asshown by the filters on lines 10 and 30 in FIG. 3 (where,illustratively, N₁=16 and N₂=12) and the resulting gap region 40. Thisgap is filled with a transition analysis filter V_(1,2)(e^(j(ω+ω) ⁰ ⁾)having a window ν_(1,2).

The shape of window ν_(1,2) is designed to provide for theaforementioned near perfect reconstruction of a signal (in the absenceof processing between the analysis filters and the synthesis filters).When the transition analysis filter is chosen to be subsampled at rateK₂ (and the transition synthesis filter upsampled at rate K₂) the aboveconstraint also means that the shape of window ν_(1,2) satisfies theexpression

$\begin{matrix}{{{{\frac{1}{K_{1}}{\sum\limits_{i = 0}^{n_{1} - 1}\;{{V_{1}\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N_{1}}i}})}} \right)}}^{2}}} + {\frac{1}{K_{2}}{{V_{1,2}\left( {\mathbb{e}}^{j{({\omega - \omega_{0}})}} \right)}}^{2\;}} + {\frac{1}{K_{2}}{\sum\limits_{i = n_{2}}^{N_{2} - 1}\;{{V_{2}\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N_{2}}i}})}} \right)}}^{2}}}} \approx R},} & (13)\end{matrix}$where R is a constant. This can be achieved by selecting the ν_(1,2)window to have the shape of the ν₂ at positive frequencies, and theshape of the ν₁ at negative frequencies; but the latter being scaled by√{square root over (N₁/N₂)}. That is the arrangement shown on line 30 ofFIG. 3. The physical filter arrangement is shown in FIG. 4, with filterbank 150 providing the response of line 10 in FIG. 3, filter bank 160providing the response of line 30 in FIG. 3, and filter 170 providingthe response of line 20 in FIG. 3.

In a relatively simple embodiment, the subsampling factors (K₁, K₂) andwindow bandwidths (2π/N₁, 2π/N₂) are selected so that N₁/K₁=N₂/K₂=R, andthe “break-over” frequency, ω₀, is chosen to satisfy the condition2πn ₁ /N ₁=2πn ₂ /N ₂=ω₀;  (14)where n₁ and n₂ are integers. In FIG. 3, for example, N₁=16, N₂=12, n₁=4and n₂=3. Filter bank 150, in accordance with the above, includesfilters V₁(ze^(−j2πi/N) ¹ ), where i=0,1,2, . . . , n₁−1, . . . ,N₁−n₁+1, N₁−n₁+2, . . . , N₁−1, and filter bank 170 includes filtersV₂(ze^(−j2πi/N) ² ), where i=n₂+1, n₂+1, . . . , N₂−n₂−1.

Based on the above, one can conclude that in order to have nearlyperfect reconstruction in a nonuniform filter bank that comprisesuniform filter bank sections, the window of each of the uniform filterbanks should have sufficiently high attenuation for ω greater than 2πdivided by the respective value of K of the filter, and alsosufficiently high attenuation for ω greater than 2π divided by therespective value of N of the filter. The attenuation for ω>2π/K controlsaliasing, while the high attenuation for ω>2π/N facilitates design oftransition filters for attaining nearly perfect reconstruction with thenonuniform filter bank. Roughly speaking, the attenuation at ω=2π/N, andtwice that attenuation at ω=2π/K, expressed in decibels, have acomparable effect on the error in the approximation to equation (13).

To design a window that has a high attenuation in the band π/N to π, onecan simply impose the requirement that the integral of the energy inthat band should be minimized. As indicated above, some spilling ofenergy close to π/N is permissible, but it is considered important thatthe attenuation that is farther removed from π/N should be large. Toachieve this result, it is suggested that the function to be minimizedmight be one that accentuates higher frequencies. For example, one mightchoose to minimize the weighted integral

$\begin{matrix}{E_{N} = {\int_{\pi\text{/}N}^{\pi}{{{V\left( {\mathbb{e}}^{j\omega} \right)}}^{2}\omega^{3}{{\mathbb{d}\omega}.}}}} & (15)\end{matrix}$This energy function is given by the quadratic formE _(N)=ν^(T) Tν,  (16)where ν is the window (time) function in the form of a column vector andthe matrix T comprises the elements:

$\begin{matrix}{\lbrack T\rbrack_{i,j} = {\int_{\pi\text{/}N}^{\pi}{\omega^{3}{\cos\left( {\left( {{\mathbb{i}} - j} \right)\omega} \right)}\ {{\mathbb{d}\omega}.}}}} & (17)\end{matrix}$The window design requires minimization of the quadratic form inequation (16) under the power complementarity condition which, expressedin the time domain, takes the form of the following set of quadraticconstraints:

$\begin{matrix}{{\sum\limits_{n}\;{{v\lbrack n\rbrack}{v\left\lbrack {n + {{\mathbb{i}}\; N}} \right\rbrack}}} = {{\delta\lbrack{\mathbb{i}}\rbrack}.}} & (18)\end{matrix}$This is a perfectly valid design approach, but it can be numericallyvery extensive and hard to implement for long windows. A simpler andfaster approach to window design is found in an article by the inventorherein, Z. Cvetkovic, “On Discrete Short-Time Fourier Analysis,” IEEETransactions on Signals Processing, Vol. 48, No. 9, September 2000, pp.2628-2640, which is hereby incorporated by reference, and is brieflydescribed below. This simpler design approach represents a window usinglinear combination of discrete prolate spheriodal sequences that areeigen vectors of matrices S_(L)(α), with matrix elements:[S _(L)(α)]_(i,j)=sin((i−j)α)/(i−j)π, 1≦i, j≦L, α=π/N.  (19)Given a column vector ν, the quadratic form ν^(T)S_(L)(α)ν gives energyof ν in the frequency band (0, π/N). The eigen vectors ρ₀, ρ₁, . . . ,ρ_(L−1) of S_(L)(α) are orthogonal, and corresponding eigenvalues λ₀,λ₁, . . . , λ_(L−1) are distinct and positive. Sorting the eigenvectorsso that λ₁>λ_(i+1), a window to be designed is constructed from a linearcombination for the first L/N+k+1 eigenvectors,

$\begin{matrix}{v = {\sum\limits_{i = 0}^{{L\text{/}N} + k}\;{a_{i}\rho_{i}}}} & (20)\end{matrix}$where we take k to be a number between 5 and 10. The constraints ofequation (18) translate into the following set of constraints in theexpansion coefficients,

$\begin{matrix}{{{\sum\limits_{l,{m = 0}}^{{L\text{/}N} + k}\;{c_{lm}^{(k)}a_{l}a_{m}}} = {\delta\lbrack k\rbrack}},{k = 1},2,\ldots\mspace{14mu},{{L\text{/}N} - 1}} & (21)\end{matrix}$where

$\begin{matrix}{c_{lm}^{(k)} = {\sum\limits_{n}\;{{\rho_{l}\lbrack n\rbrack}{{\rho_{m}\left\lbrack {n + {kN}} \right\rbrack}.}}}} & (22)\end{matrix}$The design then amounts to finding the minimum of the quadratic forma^(T)T_(a)a under the constraints of equation (21), wherea=[α ₀ . . . α_((L/N+k))]^(T),   (23)T_(a)=r^(T)Tr, and r is the matrix of the first L/N+k+1 eigenvectorsρ_(i),r=[ρ ₀ρ₀ . . . ρ_(L/N+k)].  (24)

A transition window ν_(1,2) for joining a uniform section of anN₁-channel filter bank based on a window ν₁ at low frequencies with auniform section of an N₂-channel filter bank based on a window ν₂ athigh frequencies is designed by approximating, as closely as possible,the frequency response of ν₁ at negative frequencies, and the frequencyresponse of ν₂ at positive frequencies. This amounts to a minimizingenergy function E_(tr) given by

$\begin{matrix}{E_{tr} = {{\int_{\pi}^{0}{{{{V_{1}\left( {\mathbb{e}}^{j\omega} \right)} - {V_{1,2}\left( {\mathbb{e}}^{j\omega} \right)}}}^{2}\ {\mathbb{d}\omega}}} + {\int_{0}^{\pi}{{{{\beta\;{V_{2}\left( {\mathbb{e}}^{j\omega} \right)}} - {V_{1,2}\left( {\mathbb{e}}^{j\omega} \right)}}}^{2}\ {\mathbb{d}\omega}}}}} & (25)\end{matrix}$where β=√{square root over (N₁/N₂)}, and carrying out the design processas discussed above.

It is noted that the choice of ω₀ is not completely unconstrainedbecause, if one wishes the relationship

$\omega_{0} = \frac{2\pi\; n_{1}}{N_{1}}$to hold, where both n₁ and N₁ are integers; and this is particularlytrue when the value of N₁ is constrained by other design preferences.This, however, is not an absolute limitation of the design approachdisclosed herein because as long as a gap is left between filter banks(like gap 40) and a transition filter is designed that meets therequirements of equation (13), near perfect reconstruction performanceis attainable.

The above disclosed an arrangement where the frequency band of interestis divided into two bands, with each of the two bands being handled byone filter bank, and with a single transition filter between the twofilter banks. It should be realized that this design approach could beeasily extended to a non-uniform filter bank that includes any number ofuniform filter banks, with each uniform filter bank segments having alower cutoff frequency, ω^(i) _(lower), and an upper cutoff frequency,ω^(i) _(upper) and the number of filters in the bank being dictated bythe two cutoff frequencies and the desired bandwidth, 2π/N_(i).

The above disclosed an approach for creating a filter bank that performsa non-uniform decomposition of a signal having a given bandwidth, bymeans of an example where two different filter bank sections are joinedusing a transition filter. A more general non-uniform filter bank can becreated by joining non-overlapping sections of any number of uniformfilter banks, using a plurality of transition filters. Many variationscan be incorporated by persons skilled in the art based on the disclosedapproach, without departing from the spirit and scope of this invention.For example, assume that three bands are desired, with “break-over”frequencies ω₀ and ω₁. A set of constants can be selected so that2πn₁/N₁=2πn_(2,1)/N₂=ω₀ and 2πn_(2,2)/N₂=2πn₃/N₃=ω₁; filters V₁, V₂, V₃can be selected, together with filters V_(1,2) and V_(2,3) can bedesigned, as described above.

1. An arrangement comprising: a collection of filters indexed by i,where i=0, 1, 2, . . . n₁, wherein n₁ is an integer less than N₁, eachhaving an input that is responsive to a different signal, where filter ihas the transfer function${{\overset{\_}{V}}_{1}\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N_{1}}i}})}} \right)},$ and V is a time-reversed version of a filter V having high attenuationfor frequencies ω>2π/N that is subject to the constraint that${\sum\limits_{i = 0}^{N_{1} - 1}\;{{V_{1}\left( {\mathbb{e}}^{j{({\omega + {\frac{2\pi}{N_{1}}i}})}} \right)}}^{2}} = N_{1}$ for all ω∈(−π, π); and a combiner for adding output signals of saidfilters to form an output signal of said arrangements.
 2. Thearrangement of claim 1 further comprising a second collection offilters, indexed by i, where i=n₂+1, n₂+2, . . .${\frac{N_{2}}{2} - n_{2}},$ wherein n₂ is an integer less than N₂, eachhaving an input that is responsive to a different signal, where filter jhas the transfer function${{\overset{\_}{V}}_{2}\left( {\mathbb{e}}^{j{({\omega - {\frac{2\pi}{N_{2}}i}})}} \right)},$and V is a time-reversed version of a filter V having high attenuationfor frequencies ω>2π/N that is subject to the constraint that${\sum\limits_{i = 0}^{N_{2} - 1}\;{{V_{2}\left( {\mathbb{e}}^{j{({\omega + {\frac{2\pi}{N_{2}}i}})}} \right)}}^{2}} = N_{2}$for all ω∈(−π, π); and said combiner also adds output signals of saidfilters in the second collection of filters.
 3. A non-uniform filterbank comprising: a B plurality of filter bank sections, each spanning achosen non-overlapping frequency band, where filter bank section jincludes a filter V_(j) of bandwidth 2π/N_(j) and n_(j) of itsmodulates, each responsive to a different signal and each developing anoutput signal, with said n_(j) output signals combined to form an outputof section j; and at least one transition filter for joining twoadjacent filter bank sections j=k and j=k+1, that is a modulate of afilter V_(k,k+1) with a shape of V_(k+1) at positive frequencies and ashape of V_(k) at negative frequencies, with the latter scaled by√{square root over (N_(k)/N_(k+1))}.